In principle, the information is gained at principle at the price of disturbing the system through the interaction and the system will be projected onto certain eigenstates in quantum mechanics. However, in weak measurement, the disturbance to the system is reduced at the cost of a similar reduction in the amount of information provided by the measurement[1]}. In this case, the probe shifts, averaged over many measurement repetitions, have been amplified by the weak value \(\rm A_w\)  of the observable \(\hat{A}\) : \(A_{w}:={\langle {\Phi _{f}}|\hat{A}|{\Phi _{i}}\rangle }/{\langle {\Phi _{f}}|{\Phi _{i}}\rangle },\)  where \(|{\Phi _{i}}\rangle \)  and \(|{\Phi _{f}}\rangle \)  are the pre-selected and post-selected states of the system. It has been suggested that a standard interferometer greatly outperforms weak measurement in a scenario involving a purely real weak value, but the measurement of a phase using a purely imaginary weak value can outperform a standard interferometer[2]}. Hence, anomalous works[3]}, [4]}, [5]}, [6]} of amplifying the detector signals based on weak measurement with a purely imaginary weak value. So far, the weak measurements involving a plural weak value have been seldom reported.
